How to Prove Two Sets are Equal: (Prove if A x C = B x C then A = B)

TL;DR

If a cross C is equal to B cross C and a equals B, then a is equal to B.

Transcript

let a b and c be sets and C not equal to the empty set prove that if a cross C is equal to B cross C and a is equal to B proof so this is an if-then statement so I start by assuming that this piece here is true and then we have to show that this piece here is true so we'll start by writing suppose that a cross C is equal to B cross C and now we hav... Read More

Key Insights

• 😵 The proof relies on the assumption that a cross C is equal to B cross C.
• 😫 The non-emptiness of set C is necessary to establish the existence of certain elements.
• 👍 The method of double inclusion is used to prove that a is a subset of B and vice versa.

Questions & Answers

Q: What does the proof assume initially in order to proceed?

The proof assumes that a cross C is equal to B cross C.

Q: How does the non-emptiness of C contribute to the proof?

The non-emptiness of C ensures the existence of an element C in capital C, which is necessary to establish the relationship between elements in a cross C and B cross C.

Q: How does the proof show that every element in a is in B?

By assuming an element a in capital a, the proof demonstrates that a comma C, where C is an element in capital C, belongs to B cross C. Therefore, a must be an element in capital B.

Q: How is it proven that every element in B is in a?

Similar to the previous case, the proof assumes an element B in capital B and shows that B comma C, where C is an element in capital C, belongs to a cross C. Thus, B must be an element in capital a.

Summary & Key Takeaways

• The video presents a proof concerning the equality of two cross products, given that the sets a and B are equal.

• The proof uses the method of double inclusion, showing that a is a subset of B and vice versa.

• By demonstrating that every element in a is in B and vice versa, it concludes that a is equal to B.